Let Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→ℝr and define Xt:Cr[0,t]→ℝ(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<⋯<tn=t. In this paper, using a simple formula for the conditional expectations of the functions on Cr[0,t] given Xt, we evaluate the conditional analytic Feynman integral Eanfq[Ft∣Xt] of Ft given by where θ(s,⋅) are the Fourier–Stieltjes transforms of the complex Borel measures on ℝr, and provide an inversion formula for Eanfq[Ft∣Xt]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Eanfq[Ft∣Xt] and a probability distribution on ℝr when Xt(x)=(x(0),x(t)).

The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.

It is shown that the dual of the space Cp(I) of all real-valued continuous functions on the closed unit interval with the pointwise topology, when equipped with the Mackey topology, is a non K-analytic but weakly analytic locally convex space.

A reverse iterated function system is defined as a family of expansive maps {T1,T2,…,Tm} on a uniformly discrete set . An invariant set is defined to be a nonempty set satisfying F=⋃ j=1mTj(F). A computation method for the dimension of the invariant set is given and some questions asked by Strichartz are answered.

Functions in the meromorphic Besov, Qp and related classes are characterized in terms of double integrals of certain oscillation quantities involving chordal distances. Some of the results are analogous to the corresponding results in the analytic case.

Let (R,𝔪) be a commutative Noetherian local ring, let I be an ideal of R and let M and N be finitely generated R-modules. Assume that , . First, we give the formula for the attached primes of the top generalized local cohomology module HId+n(M,N); later, we prove that if Att(HId+n(M,N))=Att(HJd+n(M,N)), then HId+n(M,N)=HJd+n(M,N).

In this note, we give an example of a Hausdorff, countable monotone Lindelöf space which is not metrizable, which gives a negative answer to a question raised by Levy and Matveev.

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this result is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply our theorem to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one.

The duality proof of sampling localization given by Loy, Newbury, Anderssen and Davies in 2001 contains an oversight, as the classes of functions chosen do not assume the compact support. Here, it is shown how a minor change to the argument there yields a precise conclusion.

Using some techniques of perturbation theory for Banach space complexes, we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation under small (with respect to the gap topology) perturbations with linear relations.

We establish a link between regular subgroups of the affine group and radical circle algebras on the underlying vector space.

In this article we show that finite loops with nilpotent inner mapping groups are centrally nilpotent.

In this paper we consider the problem of finding standing waves – solutions to nonlinear Schrödinger equations with vanishing potential and sign-changing nonlinearities. This involves searching for solutions of the problem (1)We show that the problem has a solution, and the maximum point of the solution is concentrated on a minimum point of some function as ε→0.

In this paper, a general construction of a skew-product dynamical system, for which the skew-product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes-type systems) are investigated. It is shown that the Milnes-type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of Ellis groups of any skew-product dynamical system is calculated.

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.

In this note we study the zeros of solutions of differential equations of the form u′′+pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.

Let E be a totally real set on a Stein open set Ω on a complete noncompact Kähler manifold (M,g) with nonnegative holomorphic bisectional curvature such that (Ω,g) has bounded geometry at E. Then every function f in a Cp class with compact support on Ω and -flat on E up to order p−1,p≥2 (respectively, in a Gevrey class of order s>1, with compact support on Ω and -flat on E up to infinite order) can be approximated on compacts subsets of E by holomorphic functions fk on Ω with degree of approximation equal k−p/2 (respectively, exp (−c(s)k1/2(s−1)) ).